I'd like to understand a particular random graph process that I'll describe below. I don't know if it is difficult or elementary, any pointers would be helpful.
For a given set $\{1,\dots,n\}$ of colors, we consider digraphs such that every edge has a color, and every vertex has at most $1$ outgoing edge of each color.
We can build such a graph on a fixed vertex set $V$ as follows: pick a starting vertex $v_0\in S$ at random. From then on, let the current vertex be $v_i$, pick a color at random, and pick $v_{i+1}$ uniformly at random from the vertices that could still have an edge to $v_i$ of that color (including those that already have such an edge), and add that edge if it doesn't already exist. Then move to $v_{i+1}$.
I'd like to understand the graph that this process constructs. For example, for any digraph we can ask about the standard deviation of the indegree of vertices in the graph. What is the expected value of this standard deviation for a graph constructed as above on a set $V$, with $n$ colors, after $k$ steps. This is just an example: I'd like to understand as much as possible about the constructed graph.